Define: $Z(x)\iff trs(x) \land \forall y \in x \, (\exists k (y= \{k\}) \lor (y \subset x \land trs(y) \land \forall z \in y \, (\{z\} \in y))$
Where: $trs(y) \iff \forall a \, \forall b \, (a \in b \in y \to a \in y)$
In words: a Zermelo set is a transitive set of singletons or transitive subsets of it that are closed under singletons.
I'm assuming "Foundation scheme" here.
Now we define a Zermelo ordinal as an element of a Zermelo set.
What I call as Zermelo \von Neumann ordinal race is the following principle:
If an initial class $I$ of Zermelo ordinals is bounded in cardinality by a class of von Neumann ordinals that can be placed in one-to-one relation with a subclass of $I$, then there is a Zermelo ordinal bigger than all of those Zermelo ordinals, and also a von Neumann ordinal bigger than all of those placed von Neumanns.
Where: "class bounded in cardinality by a class" means that every element of the former class is smaller or equal in cardinality to an element of the latter class. An initial class, is a class closed under predecessor relation.
Now if we add this principle to $\sf Z - Infinity + \forall x \exists \alpha: x \in V_\alpha$, would that get us to $\sf ZF$?